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Local Resolvent Estimates for <Emphasis Type="Italic">N</Emphasis>-body Stark Hamiltonians

Authors:	Tadayoshi Adachi

Institution:	(1) Department of Mathematics, Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe-shi, Hyogo 657-8501, Japan

Abstract:	For an N-body Stark Hamiltonian $H = -\Delta/2 - \|E\|z + V$ , the resolvent estimate $\\|\langle x\rangle^{-\sigma'-1/4}(H - \zeta)^{-1}\langle x\rangle^{-\sigma'-1/4}\\|_{\boldsymbol{B}(L^2)}\le C$ holds uniformly in $\zeta \in \boldsymbol{C}$ with Re $\zeta \in I$ and Im $\zeta \ne 0$ , where $\sigma' > 0$ , and $I \subset \boldsymbol{R}$ is a compact interval. This estimate is well known as the limiting absorption principle. In this paper, we report that by introducing the localization $\tilde{q}_0(x) = \sqrt{1-z/\langle x\rangle}$ in the configuration space, a refined resolvent estimate $\\|\langle x\rangle^{-1/4}\tilde{q}_0(x)(H - \zeta)^{-1}\tilde{q}_0(x)\langle x\rangle^{-1/4}\\|_{\boldsymbol{B}(L^2)} \le C$ holds uniformly in $\zeta \in \boldsymbol{C}$ with Re $\zeta \in I$ and Im $\zeta \ne 0$ . Dedicated to Professor Hideo Tamura on the occasion of his 60th birthday

Keywords:	resolvent estimate N-body Stark Hamiltonian limiting absorption principle
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